A simple and short answer I submit: God's Holy Bible--The Word-- tells us that we are to give up, don't participate, don't desire or succumb to "the ways of this world" to partake of it's pleasures and desires, or cares of this world, for they lead only to a path of destruction. He tells us to not sacrifice ourselves and all that goes with our each being IN this world, but instead to sacrifice our human desires so as to NOT follow the devil's many, many temptations. God wants and expects His children to instead sacrifice for "the cause of Christ". On point is this directive from God's Word:...be ye IN the world, but NOT! OF! this world, for the devil is OF this world! "We are here temporarily by being IN this world, for our eternal home is with the Lord of Hosts, our God and creator in Heaven"(Romans 12).
"Praise God we have a Savior and His name is Jesus Christ!" --Andy--

Human beings are categorized as a living creatures; everything that has the breath of life in it (Gen. 1: 30; 2: 7).
As believers and followers of Jesus, all of us are given a role to play to live our life on this planet earth as Salt and Light (Mat. 5: 13-16).
So, wherever we are, what ever we do, we bring blessings to our neighbors and our environments by becoming hearers and doers of the Word of God.
This is how we can be a living sacrifice.

Man has an incurable disease called sin in the flesh. John 1:8 says, if we say we have no sin we are liars and do not the truth. If we could see our sinfulness from Gods viewpoint we could not stand ourselves and would probably die on the spot.
Our trouble is we try to make the flesh good instead of walking by faith. Paul said we are not to try to fix ourselves but to wait for the spirit of God by faith to cleanse us from sin. When we got saved by faith we are to continue in that faith and wait for the righteousness that comes by faith.
No one can live the Christian life because of our old nature that we all have. There is nothing we can do to eradicate it, Paul told the Galatians that they were trying to live under the law instead of grace. They had severed themselves from grace by trying to live under the law and had lost the blessing of trusting in Christ alone by faith, and were now trying to live by sight and not walking in the spirit. When we try to obey the law in the flesh we fail because the flesh can not please the spirit.
We are to put our faith in Christ alone by trusting in His grace and He will cleanse us from all unrighteousness. We can not do the work of the Holy Spirit in us. Our self effort just places our trust in our selves instead of walking in the spirit. We are to wait for righteousness that comes by faith. The flesh can not make the flesh good.
In Galatians 3:3 Paul says, " Are ye so foolish? Having begun in the spirit are ye now made perfect by the flesh?"
So the answer to being a living sacrifice is to realize we are sinners saved by grace and we sacrifice our old nature to God so that we can walk in the spirit instead of the flesh. We walk by faith in Christ to do the work in us and leave ourselves alone. Then He can become our righteousness by faith.
Jesus said a leopard can not change its spots. The flesh can not make the flesh good. It is by faith that we wait for the righteousness that only God can do in us. Not our self effort.

Here is my opinion as regards what it means to be a living sacrifice...
Before I proceed, there are few things we need to understand as believers in order to fully grasp what a living sacrifice is and what it takes to be one for the Lord Jesus Christ.
Firstly, we must bear in mind that there is/was a dead sacrifice due to the fact that the living sacrifice is now in existence. This was brought about by the Lord Jesus Christ through the Holy Spirit.
Secondly, we are not qualified to be a living sacrifice on our own; but God must qualify us through Jesus Christ and the power of Holy Spirit to be one.
And thirdly, in the old testament, the sacrifice on the altar is/was "dead" animal and its fragrance is/was just once after the fire consumed it. But in the new testament, God desires a living sacrifice with a continuous fragrance seasoned with the fire of Holy Spirit for the Lord.
Now, what does it mean to be a living sacrifice? Casting our minds back a bit to the introduction (chapter one) of the letter, you will notice that it was not written to unbelievers but to believers who are Saints - Romans 1:6,7,8. Becoming a living sacrifice, you must first understand that the letter is meant for you. It's just a matter of time and space, you could have been in Rome where it was first read. God counted us worthy to have the same letter in this dispensation.
A typical example of a living sacrifice was demonstrated in old testament between Abraham and Isaac. Isaac wasn't dead, he knew what his father was about to do. He has no will for himself any longer but that of his father must be fulfilled. God suspended it. God blessed Abraham and Isaac for that boldness and act of faith. Genesis 22:1-19. God did it Himself with Jesus Christ right on the cross at Calvary.
The relationship between God and Jesus Christ through the Holy Spirit in Him(Jesus) needed to be passed across to all God's children. Just as we saw in Isaac, and now in Jesus Christ as they have no personal wills but that of their fathers - Abraham and God respectively. We have received same abundance grace through the Holy Spirit to do exactly the same thing; fulfilling the desire of God in all things we do in love from a pure heart,from a good conscience, and from sincere faith.
A living sacrifice is someone who believes in the saving grace of God in Christ Jesus. This sacrifice becomes new and living through the indwelling of the Holy Spirit in him/her.
In the letter of the apostle Paul to the Ephesus (2:1) we knew that we were once dead - in trespasses and sins. But made alive through the gospel of Jesus Christ. Jesus said it Himself that a time will come and now is the time that dead (sinners and unbelievers) will hear Him and become a living. John 5:25,26. The moment we become living through gospel, we can be offered/given to God as a living sacrifice not as dead any longer. And our fragrance on the altar will be continuous by the burning of Holy Spirit as fire in our spirit.
Notice this, your fragrance as a living sacrifice to God might create a double effects. To God and those who are saved - sweet fragrance and to those who are perishing an aroma of death. 2 Corinthians 2:14,15,16.
Thanks be to God for Christ Jesus our passover lamb has been slain. Amen. 1 Corinthians 5:7. Walk in this consciousness and reckon yourself through faith in Christ Jesus who qualified us as a living sacrifice, holy, and acceptable unto God the Father through the power of the Holy Spirit at work in us. Amen.
Peace be unto all Saints in Christ Jesus our Lord and Saviour. Amen

God wants us to offer ourselves, not animals, as living sacrifices, daily laying aside our own desires to follow him, putting all our energy and resources at his disposal, and trusting him to guide us. We do this out of gratitude, that our sins have been forgiven.